Consider the following linear system
Observe that since we know the number of equations and variables, the important information about this system are the number multiplying the variables and those ones on the right hand side of each equation. Thus there is a more compact way of encoding all the information needed about a linear system. We encode all the coefficients into a matrix called the matrix of coefficients
If we want to capture the whole information we encode it into the augmented matrix
Seeing a matrix as a linear system
The above representation goes both ways. Consider the matrix below
it represents the following linear system
Rodrigo Ribeiro 